A general thermodynamics-triggered competitive growth model to guide the synthesis of two-dimensional nonlayered materials

Two-dimensional (2D) nonlayered materials have recently provoked a surge of interest due to their abundant species and attractive properties with promising applications in catalysis, nanoelectronics, and spintronics. However, their 2D anisotropic growth still faces considerable challenges and lacks systematic theoretical guidance. Here, we propose a general thermodynamics-triggered competitive growth (TTCG) model providing a multivariate quantitative criterion to predict and guide 2D nonlayered materials growth. Based on this model, we design a universal hydrate-assisted chemical vapor deposition strategy for the controllable synthesis of various 2D nonlayered transition metal oxides. Four unique phases of iron oxides with distinct topological structures have also been selectively grown. More importantly, ultra-thin oxides display high-temperature magnetic ordering and large coercivity. MnxFeyCo3-x-yO4 alloy is also demonstrated to be a promising room-temperature magnetic semiconductor. Our work sheds light on the synthesis of 2D nonlayered materials and promotes their application for room-temperature spintronic devices.

The covalent binding in nonlayered materials is stronger, so the interaction force can act to next units and , +1 is theoretically related with n. As calculated in Supplementary Fig. 3, the value gets dramatically enhanced with the raise of n when n ≤ 4, while it hardly changes as n further increases (n > 4).
The interaction force of , +1 is in the vertical direction, thus the increase of vertical thickness (i.e., n) will inevitably enhance , +1 . By contrast, is determined by horizontal interaction force, which is mainly affected by the diameter of materials instead of the thickness, so the effect of n on can be ignored. Taking ε-Fe2O3 as an example here ( Supplementary Fig. 5a, b), the thickness can be largely reduced with increasing the mass of hydrates, indicating the important role of H2O as well. When the hydrates mass reaches 800 mg, the change of mass makes less effect.
After the reaction, XRD peaks of anhydrous compound appear in addition to unreacted hydrates ( Supplementary Fig. 5c), illustrating that water is released upon heating CaSO4·2H2O and is adequate for the reaction when the mass exceeds 800 mg.
Therefore, we speculate the following mechanism: On the one hand, hydrates release water vapor to react with chlorides in the gaseous atmosphere to form oxides on substrates. On the other hand, sufficient water molecules adsorb on the surface of oxides and decrease term ( Fig. 1d-h), thus 2D growth is preferred and the thickness is reduced. Moreover, we found that CaSO4·2H2O can release water at different rates by varying the temperatures ( Supplementary Fig. 5d), served as a stable source of water vapor in a much controller rate range and lower vapor pressure than liquid water without the usage of complicated setups. In other words, the top surface would possess growth steps (i.e. the incomplete growth layer) with the coverage less than 30% from normalized ADF image contrast. The number of adsorbed layers (the incomplete growth layer) varied from two to four based on the Z-contrast, and the atomic structure is similar to that of the bulk structure. In fact, the adsorbed layers can spread over hundreds of nanometers, much larger than the height (0.48~1.0 nm), indicating the lateral growth of the nonlayered 2D materials as well.
Moreover, an additional layer (denoted by blue arrows) was also observed on the topmost surface, and the arrangement is disordered compared to the below growth layers, which is likely to be the passivation layer of hydroxyl group, because there were large amounts of water vapor during the CVD synthesis process.  proportional to out-of-plane stray magnetic field emanating from the surface of the sample. Therefore, the phase difference between materials and the substrate (nonmagnetic) reflects the strength of out-of-plane magnetism of the sample. In addition, the nonuniform phase contrast of the nanoflake indicates the multiple magnetic domain structures. As is shown in Fig. 5a, b and Supplementary Fig. 27d, e, Fe3O4 and γ-Fe2O3 have stronger phase contrast with the substrate (the color of the phase image is darker). Moreover, the phase difference between the material and substrate is larger in Fe3O4 (such as ~1.5 degrees across the line of Supplementary Fig.   27d) and γ-Fe2O3 (such as ~2.2 degrees across the line of Supplementary Fig. 27e). Therefore, Fe3O4 and γ-Fe2O3 show strong out-of-plane magnetism with different magnetic domain shapes. While ε-Fe2O3 has weaker contrast with the substrate (Supplementary Fig. 27f) and smaller phase differences with 0. At thicker thickness ( Supplementary Fig. 31a), RMCD signal shows a step-like loop (indicated by the arrows) as the magnetic field sweeps upward, and finally reaches the saturation value at a high positive field. Similar step-like hysteresis phenomena were also reported in 2D CrTe2 11 , CrI3 12 , and CrBr3 13 , which may derive from the polarization of several domains that are magnetically independent when the spot size of RMCD laser is ~1 μm. When the thickness is thinner down to ~12 nm ( Supplementary Fig. 31b), step-like hysteresis is more obvious and the magnetic signal becomes weaker, illustrating the lowered magnetic exchange interaction. The large coercivity is well sustained with a little reduction (~3900 Oe for the thicker and ~3300 Oe for the thinner sample at 2 K). In addition, the multi-domain structure gets more sophisticated (the number of domain walls increases) and magnetism decreases (the phase contrast with the substrate becomes weaker) with reducing the thickness ( Supplementary Fig. 31c), which may explain the step-like hysteresis behavior of RMCD results.  We assume that the influence of substrates on can be expressed as As is shown, mica has smaller diffusion barrier energy ( ), so their resistance for edge growth is small, especially at high temperatures (growth temperature is more than 800 K), leading to smaller to promote 2D growth. Therefore, we assume that can be considered as 0 for mica substrates in our work.

Materials Structures Materials Structures
Fe3O4

1) Detailed calculation procedure of the above parameters:
, +1 is calculated by the formula , +1 = ( +1 − )/ , where the and +1 are the total free energy of slabs with n and + 1 subunits, respectively. Therefore, , +1 represents the binding energy between the subunit n and subunit + 1. In the same material, we suppose equals .
is the correction term of term. The interface adsorption will passivate the dangling bonds of the top surface, thus reducing the binding energy between the initial structure and the newly growth cluster. of water is calculated by the formula = are the total free energies of the slab with and without adsorbed H2O, respectively.  Table 2). and of the new growth cluster are assumed to be the same (supposed as unit area).
As for Fe-based oxides, the intrinsic energy differences between , +1 and + are not negative enough (most of them >-0.25 eV/Å 2 ). The facilitation of H2O passivation (large ) and mica substrate (small , Supplementary The value is negative but not particularly small, so it may form 2D structure, but the formation of ultrathin thickness without other assistance may be difficult 16 .  Fig. 6).

Supplementary Note 1: Detailed deducing process of the equations (2) and (3)
We assume a new growth cluster (the basal and lateral area are and , the number of superimposed subunits is m) combines with the initial structure (the basal and lateral area are and , the number of superimposed subunits is n) in two ways, i.e., vertically or laterally. According to the equation (1), the total free energy of the initial structure ( ) and the newly growth cluster ( ) is as follow: After vertical growth, the binding energy of the subunit at the interface will change, and the total free energy is as follows.
We suppose that the change of , +1 (influenced by n) can be ignored when n is larger than 4 (as shown in Supplementary Fig. 3). The basal contact area equals . Therefore, After lateral growth, we define the lateral contact area is . The edge energy is decreased due to the reduced lateral area, and the total free energy is as follows.
where and represent the average edge energies of the initial structure and the new cluster, respectively. Therefore,